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It seems like the equation shown in the article doesn't satisfy the condition listed on the wikipedia page. Did the author make a mistake?
by jhkim 9y ago
It seems like the equation shown in the article doesn't satisfy the condition listed on the wikipedia page. Did the author make a mistake?
- supergarfield 9y agoThe Wikipedia article gives Weierstrass's original condition, but it was later improved to just ab ≥ 1 (as noted, not very visibly, at the end of the article). However, the Nautilus article also says that "Conventional wisdom held that for any continuous curve, it was possible to find the gradient at all but a finite number of points", which is clearly not true, so I'd be cautious about its technical correctness.
- euyyn 9y agoFrom the way they try to explain it, I think they meant to say a "countable" number of points, instead of finite.
- alimw 9y agoOn what basis are you calling this assertion "clearly not true"?
- bonzini 9y agoYou can define a periodic function that goes from 0 to 1 as a straight line and then from 1 to 0 as another straight line, basically a triangle wave. It is not differentiable on an infinite (and countable) number of points: https://en.m.wikipedia.org/wiki/Triangle_wave https://en.m.wikipedia.org/wiki/Triangle_wave
- lmm 9y agoI imagine they would have been talking about functions on the interval rather than our modern sense of functions on the reals. (Of course there are plenty of examples on the interval as well, but no obvious ones)
- bonzini 9y agoTake something like arcsin sin pi/2/x(x-1), where "arcsin sin pi*x/2" is "pointy" like a triangle wave. There you have it.
- vilhelm_s 9y agoThere is a lot of discussion at this Stack Exchange question, "Is Kline right that Cauchy believed that continuous functions must be differentiable?" (https://hsm.stackexchange.com/questions/3480/is-kline-right-that-cauchy-believed-that-continuous-functions-must-be-differenti https://hsm.stackexchange.com/questions/3480/is-kline-right-...) Mathematicians didn't state this as theorem or axiom, so it's hard to pin down exactly what they might have believed; the people answering the question talk about "except at isolated points" rather than "except at a finite number of points".